A118884 Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 0011 (n,k>=0).

1, 2, 4, 8, 15, 1, 28, 4, 52, 12, 96, 32, 177, 78, 1, 326, 180, 6, 600, 400, 24, 1104, 864, 80, 2031, 1827, 237, 1, 3736, 3800, 648, 8, 6872, 7800, 1672, 40, 12640, 15840, 4128, 160, 23249, 31884, 9846, 556, 1, 42762, 63704, 22844, 1752, 10, 78652, 126480

Offset: 0

Author

Emeric Deutsch, May 03 2006

Keywords

• nonn
• tabf

Comments

Row n has 1+floor(n/4) terms. Sum of entries in row n is 2^n (A000079). T(n,0) = A008937(n+1). T(n,1) = A118885(n). Sum(k*T(n,k), k=0..n-1) = (n-3)*2^(n-4) (A001787).

Examples

			T(9,2) = 6 because we have aa0, aa1, a0a, a1a, 0aa and 1aa, where a=0011.
Triangle starts:
1;
2;
4;
8;
15, 1;
28, 4;
52, 12;
96, 32;
		

Links

• Alois P. Heinz, Rows n = 0..300, flattened

Crossrefs

Cf. A000079, A008937, A118885, A001787.

Programs

Maple

G:=1/(1-2*z+(1-t)*z^4): Gser:=simplify(series(G,z=0,23)): P[0]:=1: for n from 1 to 19 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 0 to 19 do seq(coeff(P[n],t,j),j=0..floor(n/4)) od; # yields sequence in triangular form

Mathematica

nn=12;c=0;Map[Select[#,#>0&]&,CoefficientList[Series[1/(1-2x - (y-1)x^4/ (1-(y-1)c)), {x,0,nn}],{x,y}]]//Grid (* Geoffrey Critzer, Dec 25 2013 *)

Formula

G.f.: G(t,z) = 1/[1-2z+(1-t)z^4]. T(n,k) = 2T(n-1,k)-T(n-4,k)+T(n-4,k-1) (n>=4,k>=1).